Perfect Orderings on Finite Rank Bratteli Diagrams

Abstract

Abstract. Given a Bratteli diagram B, we study the set O B of all possible orderings on B and its subset P B consisting of perfect orderings that produce Bratteli-Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set O B with product measure µ and prove that there is some 1 ≤ j ≤ k such that µ-almost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then µ-almost all orderings are imperfect.

Invariant measures for Cantor dynamical systems

“…. The Bratteli diagram defined above admits an order generating the Bratteli-Vershik homeomorphism (see [HPS92], [GPS95], or [BKY14], [BK16]). In particular, we can use the so called consecutive ordering such that X B has the unique minimal infinite path passing through the vertices 0 ∈ V n , n ≥ 0 and the unique maximal infinite path passing through the vertices n ∈ V n , n ≥ 0.…”

Bratteli–Vershik models and graph covering models

“…Bezuglyi, Kwiatkowski, and Yassawi [BKY14] considered cases in which, for finiterank Bratteli diagrams, the order of the diagram admits a homeomorphic Vershik map. They refer to such orderings as perfect.…”

Decisive Bratteli–Vershik models

“…Let us also mention that decisive ordered Bratteli diagrams have been in fact studied, for instance in [BKY14,BY16], but from a different perspective (and without using the notion of decisiveness). The authors start with a fixed Bratteli diagram B and search for orderings < such that the corresponding Vershik map can be uniquely prolonged to a homeomorphism of the whole path-space.…”